The watergun is a toy Ellie and Riley use for a water fight in the Left Behind chapter Fun and Games. Riley stole their confiscated pair back as a surprise, and winning the fight earns the Skillz trophy.
Keep pumping between bursts so you are never caught empty, and use cover in the arena to make Riley come to you.
After the fight Riley gives Ellie both waterguns, saying she won't need them once she leaves with the Fireflies.